#147616

order	512
nilpotency class	4
isoclinism	isoclinic class of #147614
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10966
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,6,11,7,12,5,10)(3,13,16,26,17,27,14,9)(4,8,18,22,19,24,20,23)(15,25,28,21,30,31,29,32);(2,8)(3,14)(5,6)(9,25)(10,23)(11,22)(12,24)(13,21)(15,28)(16,17)(18,20)(26,31)(27,32)(29,30);(1,3,7,17)(2,9)(4,15,19,30)(5,16,6,14)(8,21)(10,13)(11,27)(12,26)(18,29,20,28)(22,25)(23,31)(24,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54812);